On the invariance of irregular Hodge numbers under crepant birational equivalences

Abstract

The Batyrev--Kontsevich theorem asserts that birational Calabi--Yau varieties have the same Hodge numbers. In this article, we prove an analogue for Landau--Ginzburg models $(U,f)$, consisting of smooth quasi-projective complex varieties $U$ with regular functions $f$ on $U$. We show that the irregular Hodge numbers of the twisted de Rham cohomology $\mathrm{H}^k_{\mathrm{dR}}(U,f)$ are invariant under crepant birational equivalences.